How to Add and Subtract
Positive and Negative Numbers

Numbers Can be Positive or Negative:

Negative Numbers (−) Positive Numbers (+)
(This is the Number Line, read about Using The Number Line)

"−" is the negative sign. "+" is the positive sign

No Sign Means Positive

If a number has no sign it usually means that it is a positive number.

Example: 5 is really +5

Adding Positive Numbers

Adding positive numbers is just simple addition.

Example: 2 + 3 = 5

is really saying

"Positive 2 plus Positive 3 equals Positive 5"

 

You could write it as (+2) + (+3) = (+5)

Subtracting Positive Numbers

Subtracting positive numbers is just simple subtraction.

Example: 6 − 3 = 3

is really saying

"Positive 6 minus Positive 3 equals Positive 3"

 

You could write it as (+6) − (+3) = (+3)

Balloons and Weights

 

This basket has balloons and weights tied to it:

  • The balloons pull up (positive)
  • And the weights drag down (negative)

 

Here is what adding and subtracting positive numbers looks like:

 

You can add balloons (you are adding positive value)

the basket gets pulled upwards (positive)

 

 

You can take away balloons (you are subtracting positive value)

the basket gets pulled downwards (negative)

 

Now let's see what adding and subtracting negative numbers looks like:

 

You can add weights (you are adding negative values)

the basket gets pulled downwards (negative)

 

 

And you can take away weights (you are subtracting negative values)

the basket gets pulled upwards (positive)

 

That last one was interesting ... subtracting a negative made the basket go up

Subtracting a Negative is the same as Adding

Example: What is 6 − (−3) ?

6−(−3) = 6 + 3 = 9

Example: What is 14 − (−4) ?

14−(−4) = 14 + 4 = 18

 

We also found that taking away balloons (subtracting positives) or adding weights (adding negatives) both made the basket go down.

And Positive and Negative Together ...

Subtracting a Positive
or
Adding a Negative

is
Subtraction

Example: What is 6 − (+3) ?

6−(+3) = 6 3 = 3

Example: What is 5 + (−7) ?

5+(−7) = 5 7 = −2

 

The Rules:

It can all be put into two rules:

  Rule   Example
Two like signs become a positive sign +(+) 3+(+2) = 3 + 2 = 5
−(−) 6−(−3) = 6 + 3 = 9
       
Two unlike signs become a negative sign +(−) 7+(−2) = 7 2 = 5
−(+) 8−(+2) = 8 2 = 6
       

They are "like signs" when they are like each other (in other words: the same).

 

So, all you have to remember is:

Two like signs become a positive sign

Two unlike signs become a negative sign


Example: What is 5+(−2) ?

+(−) are unlike signs (they are not the same), so they become a negative sign.

5+(−2) = 5 2 = 3

Example: What is 25−(−4) ?

−(−) are like signs, so they become a positive sign.

25−(−4) = 25+4 = 29

Example: What is −6+(+3) ?

+(+) are like signs, so they become a positive sign.

−6+(+3) = −6 + 3 = -3

Start at −6 on the number line, move forward 3, and you end up at −3

Now Play With It!

  Try playing Casey Runner, you need to know the rules of positive and negative to succeed!

 

A Common Sense Explanation

And there is a "common sense" explanation:

If I say "Eat!" I am encouraging you to eat (positive)

But if I say "Do not eat!" I am saying the opposite (negative).

Now if I say "Do NOT not eat!", I am saying I don't want you to starve, so I am back to saying "Eat!" (positive).

So, two negatives make a positive, and if that satisfies you, then you are done!

 

Another Common Sense Explanation

A friend is +, an enemy is

+ + ⇒ +     a friend of a friend is my friend
+ − ⇒ −     a friend of an enemy is my enemy
− + ⇒ −     an enemy of a friend is my enemy
− − ⇒ +     an enemy of an enemy is my friend

A Bank Example

Example: You have $80 in your account right now.

Last year the bank subtracted $10 by mistake, and they want to fix it.

So the bank must take away a negative $10:

$80−(−$10) = $80 + $10 = $90

So you get $10 more in your account.

Another Example You Might Like

Ally's Points

Ally can be naughty or nice. So Ally's parents have said

"If you are nice we will add 3 points (+3).
If you are naughty, we take away 3 points (−3).
When you reach 30 Points you get a toy."

Ally starts the day with 9 Points:

 
Ally's Mom discovers spilt milk: 9−3 = 6
 
Then Dad confesses he spilt the milk and writes "undo". Mom calculates: 6−(−3) = 6+3 = 9

So if you subtract a negative, you gain points
(i.e. the same as adding points).

So Subtracting a Negative is the same as Adding


 

 
A few days later. Ally has 12 points.  
Mom adds 3 points because Ally's room is clean. 12+3 = 15
Dad says "I cleaned that room" and writes "undo" on the chart. Mom calculates: 15 − (+3) = 12
Dad sees Ally brushing the dog. Writes "+3" on the chart. Mom calculates: 12 + (+3) = 15
Ally throws a stone against the window. Dad writes "−3" on the chart. Mom calculates: 15 + (−3) = 12

See: both "15 − (+3)" and "15 + (−3)" result in 12.

So:

It doesn't matter if you subtract positive points
or add negative points,
you still end up losing points.

So, Subtracting a Positive
or
Adding a Negative

is
Subtraction

 

Try These Exercises ...

Now try This Worksheet, and see how you go.

And also try these questions: